calculus question...?

Question

Find the absolute maximum and the absolute minimum, if either exists, for the function f(x) = 80x^3 -5x^4. If a value does not exist, enter 'N'.

I came up with absolute maximum = 34560 and absolute minimum = N

Is this correct?

Answer 1

You are correct, the absolute maximum is at x=12.

f(x) = 80x^3 -5x^4

f(12)=80(12)^3-5(12)^4

f(12)=34560

and there is no min

Answer 2

f(x) = 80x³ - 5x^4

To find the critical points take the derivative and set equal to zero.

f'(x) = 240x² - 20x³ = 0
20x²(12 - x) = 0
x = 0, 12 are critical points.

To determine the nature of the critical points take the second derivative.

f''(x) = 480x - 60x² = 60x(8 - x)
f''(0) = 0 implies point of inflection
f''(12) < 0 implies relative maximum

Since we can see it is an even powered function with a negative coefficient given to the largest power of x, it has an absolute maximum and no absolute minimum. Therefore the relative maximum at x = 12 is also the absolute maximum.

f(x) = 80x³ - 5x^4 = 5x³(16 - x)
f(12) = 5(12³)(16 - 12) = 20(1728) = 34560

The absolute maximum is 34,560 at x = 12.
There is no absolute minimum. So 'N'.